How white draws this endgame? Some general endgame tips, please!

Firstly, you must notice this was a 1-minute bullet game, and I, to be honest, play bullet horribly. But I am interested how to draw such endings in bullet (and in other types of chess)? Should I keep rooks on? What should me my plan of drawing such king&pawn ending?

Another note: We both had more than 20s on our clocks after we exchanged rook, so I wasn't able to win on time.

You shouldn't have exchanged rooks. You should have played for h4 followed by g3 (get your king active first before he tries to cut it off). From there try not to move your pawns but exchange them when your opponent advances his pawns. When you reach a R+P vs R endgame put your rook on the third rank and make sure your king is on the file in front of the pawn and the draw is easy from there.

Can I ask what was the point of 3.Bb5? It's not really an effective pin at this point, is it? The center's still not under control and you haven't developed your Knights. In the end you just had to maneuver your Bishop down and down the board.

The main mistake, as said by others, was exchanging rooks. For one thing extra pawn usually wins in pawn endings but perhaps even more importantly pawn endings can usually be calculated to the end. Therefore you should never enter one unless you are sure about the result because if there's a win the chances are that your opponent will find it.

However, in the game black advances his pawns bit carelessly with 35... g4?? (35 f4... should win) and thanks to black's backward f-pawn you could draw with 38. Kd3! (keeping the opposition).